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Physical review. B, Condensed matter

Macroscopically doped chiral-spin-liquid state

Z. Zou, J. L. Levy, R. B. Laughlin

Published Jan 1, 1992• 14 citations
Publisher

Abstract

Our previous work on the isolated charge excitations, holons, of the chiral-spin-liquid state is generalized to include a macroscopic number of holons. We propose a specific class of wave functions for multiholon excitations exhibiting 1/2-fractional statistics. The fractional statistics are demonstrated by means of analytic continuation methods, numerical calculations, and U(1) gauge-theory techniques. A Chern-Simons term is derived in a long-wavelength effective action for the holons. The exactness of the fractional statistics follows from the quantization of the coefficient of the Chern-Simons term. This particular class of wave functions seems energetically favorable for the t-J Hamiltonian in the small-doping limit for physical J/t, while in the large-doping limit the stability of these states requires large J/t. An upper bound for the ground-state energy of the t-J Hamiltonian within the fractional-statistics basis set is obtained.

DOI
10.1103/physrevb.45.993

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On the possibility of fractional statistics in the two-dimensional t-J model at low doping

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The purpose of this work is the derivation of an effective field theory for the low-energy magnetic modes of the t-J model on a two-dimensional square lattice and for small values of doping, which is of relevance to the physics of high-temperature superconductors. In particular, we address the possibility of low-energy excitations that obey fractional spin and statistics. For temperatures where kBT is considerably smaller than the magnetic exchange coupling J and for low values of doping, the system is close to an antiferromagnetically ordered Néel state and the spin correlation length takes values significantly larger than the lattice constant a, such that a field theoretic analysis is justified. The effective model is obtained by means of a gradient expansion around the antiferromagnetically ordered reference state. Dimensional analysis shows that in (2+1) dimensions only terms up to order O(a2) in the effective action are relevant to the behaviour at large scales, whereas O(a3)-terms are marginal and higher order terms are irrelevant. Even though marginal, the O(a3)-contributions may drastically influence the properties of excitations, as they might feature a term of topological nature which endows field histories with a statistical phase factor. In fact, all field histories can be characterized as mappings from compactified spacetime, the three-sphere S3, to the order parameter space S2 and thus fall into different homotopy classes. The topological invariant characterizing these homotopy classes is given by the Hopf invariant. Consequently, if the Hopf invariant emerges in the effective field theory, the spin and statistics of low-energy excitations fractionalize. The analysis is based on a path integral representation of the t-J model which was obtained recently by means of Dirac quantization. After introducing a staggered quantization axis, the single occupancy constraint which is inherent to the t-J model can be taken into account exactly. We perform a long-wavelength, low-frequency gradient expansion of the effective action and integrate over the fermionic degrees of freedom as well as the fast-fluctuating bosonic modes. Since our derivation is based on a microscopic model, we obtain an effective action where the doping dependence of the coupling constants is made explicit.

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cond-mat.str-elCross-listedarXiv:2607.01341•Jul 3, 2026•0 citations
Our previous work on the isolated charge excitations, holons, of the chiral-spin-liquid state is generalized to include a macroscopic number of holons. We propose a specific class of wave functions for multiholon excitations exhibiting 1/2-fractional statistics. The fractional statistics are demonstrated by means of analytic continuation methods, numerical calculations, and U(1) gauge-theory techniques. A Chern-Simons term is derived in a long-wavelength effective action for the holons. The exactness of the fractional statistics follows from the quantization of the coefficient of the Chern-Simons term. This particular class of wave functions seems energetically favorable for the t-J Hamiltonian in the small-doping limit for physical J/t, while in the large-doping limit the stability of these states requires large J/t. An upper bound for the ground-state energy of the t-J Hamiltonian within the fractional-statistics basis set is obtained.
We study Gutzwiller-projected variational wavefunctions for charged, spinless holon excitations in chiral spin liquids. We find that these states describe anyons, with a statistical phase ΦsΦ_sΦs​ that is continuously adjustable between 000 and π/2π/2π/2, depending on a variational parameter. The statistical flux attached to each holon is localized to within a lattice constant. By diagonalizing the effective Hamiltonian for charge motion in our variational basis we obtain a two-holon pair state. This two-holon state has large overlap with a two-hole state whose relative wavefunction has ddd-wave symmetry and breaks time-reversal and parity.
The purpose of this work is the derivation of an effective field theory for the low-energy magnetic modes of the t-J model on a two-dimensional square lattice and for small values of doping, which is of relevance to the physics of high-temperature superconductors. In particular, we address the possibility of low-energy excitations that obey fractional spin and statistics. For temperatures where kBT is considerably smaller than the magnetic exchange coupling J and for low values of doping, the system is close to an antiferromagnetically ordered Néel state and the spin correlation length takes values significantly larger than the lattice constant a, such that a field theoretic analysis is justified. The effective model is obtained by means of a gradient expansion around the antiferromagnetically ordered reference state. Dimensional analysis shows that in (2+1) dimensions only terms up to order O(a2) in the effective action are relevant to the behaviour at large scales, whereas O(a3)-terms are marginal and higher order terms are irrelevant. Even though marginal, the O(a3)-contributions may drastically influence the properties of excitations, as they might feature a term of topological nature which endows field histories with a statistical phase factor. In fact, all field histories can be characterized as mappings from compactified spacetime, the three-sphere S3, to the order parameter space S2 and thus fall into different homotopy classes. The topological invariant characterizing these homotopy classes is given by the Hopf invariant. Consequently, if the Hopf invariant emerges in the effective field theory, the spin and statistics of low-energy excitations fractionalize. The analysis is based on a path integral representation of the t-J model which was obtained recently by means of Dirac quantization. After introducing a staggered quantization axis, the single occupancy constraint which is inherent to the t-J model can be taken into account exactly. We perform a long-wavelength, low-frequency gradient expansion of the effective action and integrate over the fermionic degrees of freedom as well as the fast-fluctuating bosonic modes. Since our derivation is based on a microscopic model, we obtain an effective action where the doping dependence of the coupling constants is made explicit.
The Chern-Simons bosonization with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ gauge field is applied to the two-dimensional $t\ensuremath{-}J$ model in the limit $t\ensuremath{\gg}J,$ to study the normal-state properties of underdoped cuprate superconductors. We prove the existence of an upper bound on the partition function for holons in a spinon background, and we find the optimal spinon configuration saturating the upper bound on average---a coexisting flux phase and s+ids+ids+id-like resonating-valence-bond state. After neglecting the feedback of holon fluctuations on the U(1) field B and spinon fluctuations on the SU(2) field VVV, the holon field is a fermion and the spinon field is a hard-core boson. Within this approximation we show that the B field produces a \ensuremath{\pi} flux phase for the holons, converting them into Dirac-like fermions, while the VVV field, taking into account the feedback of holons produces a gap for the spinons vanishing in the zero-doping limit. The nonlinear-\ensuremath{\sigma} model with a mass term describes the crossover from the short-ranged antiferromagnetic (AF) state in doped samples to long-range AF order in reference compounds. Moreover, we derive a low-energy effective action in terms of spinons, holons and a self-generated U(1) gauge field. Neglecting the gauge fluctuations, the holons are described by the Fermi-liquid theory with a Fermi surface consisting of four ``half-pockets'' centered at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) and one reproduces the results for the electron spectral function obtained in the mean-field approximation, in agreement with the photoemission data on underdoped cuprates. The gauge fluctuations are not confining due to coupling to holons, but nevertheless yield an attractive interaction between spinons and holons leading to a bound state with electron quantum numbers. The renormalization effects due to gauge fluctuations give rise to non-Fermi-liquid behavior for the composite electron, in certain temperature range showing the linear in T resistivity. This formalism provides a new interpretation of the spin gap in the underdoped superconductors (mainly due to the short-ranged AF order) and predicts that the minimal gap for the physical electron is proportional to the square root of the doping concentration. Therefore the gap does not vanish in any direction. All these predictions can be checked explicitly in experiment.
The extended t - J model, which has both transfer integral and exchange interaction between next-nearest neighbors besides those between nearest neighbors, has been investigated by the mean field approximation in the slave-boson scheme. Especially, the staggered flux state and the chiral spin state have been investigated. For the latter state, anyons are derived by use of the result of the Chern-Simons term for the doped case. It is shown that the prefactor of the Chern-Simons term and thus the fractional statistics depend on the degree of doping.
Chiral spin liquids realize a topological state whose universal response is a fractional spin Hall conductance νsν_sνs​. The three quantities that determine this response, the integer Chern number of the fractionalized spinons, the level of the emergent Chern--Simons gauge field, and the physically measured spin pump, are related but distinct, and their relation is often stated only schematically. Here we derive it from a single object: the parity-odd determinant of a gapped Dirac cone on a spatial cylinder, resummed exactly to all orders in the compact holonomy. This determinant fixes the map from spinon topology to measurable response, and proves that finite-size corrections to the topological pump are strictly exponential, with no universal 1/L1/L1/L term. We test the resulting predictions on the kagome chiral spin liquid at three independent levels: the exact one-loop field theory, a parton band-structure calculation (C=−1C=-1C=−1, converging exponentially over cylinders four to twelve sites wide), and an interacting density-matrix renormalization group flux pump on the explicitly chiral JJJ--JχJ_χJχ​ Hamiltonian (νs=−0.500±0.011ν_s=-0.500\pm0.011νs​=−0.500±0.011). All three agree with the analytic prediction without adjustable parameters, providing a fully quantitative bridge between microscopic topology and observable fractional response.