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arXiv:1010.6094 · Physical Review B

Interactions and the θ term in one-dimensional gapped systems

Michael Mulligan

Published May 17, 2011• 5 citations
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Abstract

We study how the $\ensuremath{\theta}$ term is affected by interactions in certain one-dimensional gapped systems that preserve charge conjugation, parity, and time-reversal invariance. We exploit the relation between the chiral anomaly of a fermionic system and the classical shift symmetry of its bosonized dual. The vacuum expectation value of the dual boson is identified with the value of the $\ensuremath{\theta}$ term for the corresponding fermionic system. Two (related) examples illustrate the identification. We first consider the massive Luttinger liquid and find the $\ensuremath{\theta}$ term to be insensitive to the strength of the interaction. Next we study the continuum limit of the Heisenberg XXZ spin-1/2 chain, perturbed by a second nearest-neighbor spin interaction. For a certain range of the XXZ anisotropy, we find that we can tune between two distinct sets of topological phases by varying the second nearest-neighbor coupling. In the first we find the standard vacua at $\ensuremath{\theta}=0,\ensuremath{\pi}$, while the second contains vacua that spontaneously break charge conjugation and parity with fractional $\ensuremath{\theta}/\ensuremath{\pi}=1/2,3/2$. We also study quantized pumping in both examples following recent work.

DOI
10.1103/physrevb.83.205110

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We study how the \ensuremathθ\ensuremath{\theta}\ensuremathθ term is affected by interactions in certain one-dimensional gapped systems that preserve charge conjugation, parity, and time-reversal invariance. We exploit the relation between the chiral anomaly of a fermionic system and the classical shift symmetry of its bosonized dual. The vacuum expectation value of the dual boson is identified with the value of the \ensuremathθ\ensuremath{\theta}\ensuremathθ term for the corresponding fermionic system. Two (related) examples illustrate the identification. We first consider the massive Luttinger liquid and find the \ensuremathθ\ensuremath{\theta}\ensuremathθ term to be insensitive to the strength of the interaction. Next we study the continuum limit of the Heisenberg XXZ spin-1/2 chain, perturbed by a second nearest-neighbor spin interaction. For a certain range of the XXZ anisotropy, we find that we can tune between two distinct sets of topological phases by varying the second nearest-neighbor coupling. In the first we find the standard vacua at \ensuremathθ=0,\ensuremathπ\ensuremath{\theta}=0,\ensuremath{\pi}\ensuremathθ=0,\ensuremathπ, while the second contains vacua that spontaneously break charge conjugation and parity with fractional \ensuremathθ/\ensuremathπ=1/2,3/2\ensuremath{\theta}/\ensuremath{\pi}=1/2,3/2\ensuremathθ/\ensuremathπ=1/2,3/2. We also study quantized pumping in both examples following recent work.
We study how the θ [theta] term is affected by interactions in certain one-dimensional gapped systems that preserve charge conjugation, parity, and time-reversal invariance. We exploit the relation between the chiral anomaly of a fermionic system and the classical shift symmetry of its bosonized dual. The vacuum expectation value of the dual boson is identified with the value of the θ [theta] term for the corresponding fermionic system. Two (related) examples illustrate the identification. We first consider the massive Luttinger liquid and find the θ [theta] term to be insensitive to the strength of the interaction. Next we study the continuum limit of the Heisenberg XXZ spin-1/2 chain, perturbed by a second nearest-neighbor spin interaction. For a certain range of the XXZ anisotropy, we find that we can tune between two distinct sets of topological phases by varying the second nearest-neighbor coupling. In the first we find the standard vacua at θ=0,π [theta = 0, pi], while the second contains vacua that spontaneously break charge conjugation and parity with fractional θ/π=1/2,3/2 [0/pi = 1/2, 3/2]. We also study quantized pumping in both examples following recent work.
Semiclassical approach has been developed for the one-dimensional interacting fermion systems. Starting from the incommensurate spin density wave (SDW) mean field state for the repulsive Hubbard model in 1D, the non-Abelian bosonized Lagrangian describing the spin-charge separation is obtained. The Berry phase term is derived from the chiral anomaly, and we obtain the massless Tomonaga-Luttinger liquid in the single chain case while the spin gap opens in the double-chain system. This approach offers a new method to identify the strong-coupling fixed point, and its relation to the Abelian bosonization formalism is discussed on the spin gap state. The generalization to higher dimensions is also discussed.
The low energy behavior of a huge variety of one-dimensional interacting spinful fermionic systems exhibits spin-charge separation, described in the continuum limit by two sine-Gordon models decoupled in the charge and spin channels. Interaction is known to induce, besides the gapless Luttinger liquid phase, eight possible gapped phases, among which are the Mott, Haldane, charge-/spin-density, and bond-ordered wave insulators, and the Luther Emery liquid. Here we prove that some of these physically distinct phases have nontrivial topological properties, notably the presence of degenerate protected edge modes with fractionalized charge/spin. Moreover, we show that the eight gapped phases are in one-to-one correspondence with the symmetry-protected topological (SPT) phases classified by group cohomology theory in the presence of particle-hole symmetry P. The latter result is also exploited to characterize SPT phases by measurable nonlocal order parameters which follow the system evolution to the quantum phase transition. The implications on the appearance of exotic orders in the class of microscopic Hubbard Hamiltonians, possibly without P symmetry at higher energies, are discussed.
Effective field theory is a very useful technique for understanding quantum many body systems. We use this approach to study a certain class of critical quantum spin-1 chains and symmetry protected topological phases of matter in two spatial dimensions. In the first context we consider the SU(3) symmetric spin-1 chain with additional interactions. We use bosonization to demonstrate that the effective field theories describing the critical behaviour of these spin chains can be mapped to free compact boson conformal field theories (CFTs) with central charge c=2. We also describe how some predictions from field theory can be verified in numerical calculations using exact diagonalization (ED) and the density matrix renormalization group (DMRG) algorithm. In particular, the bosonization method gives a formula for the evolution of four Tomonaga-Luttinger liquid (TLL) parameters as a function of the lattice parameters. Using the analytic formulae for the scaling dimensions in terms of the TLL parameters and matching of lowest scaling dimensions, we numerically calculate these field-theoretic parameters and confirm that their evolution agrees with the prediction using bosonization. We also tackle aspects of the challenging problem of understanding interacting topological phases. In this context, we aim to understand the effects if interactions in certain classes of symmetry protected topological (SPT) phases of matter. We consider non-chiral SPT phases in two spatial dimensions protected by a discrete symmetry such as Z_K or Z_K x Z_K symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as examples, we demonstrate explicitly that when modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge. We also briefly discuss preliminary results of a numerical approach for simulating the 3 dimensional Landau Level problem. This includes a brief description of a potential application of variational Monte Carlo to spin-orbit coupled ab initio systems.
Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity, or on-site unitary symmetry has been given for 1D spin systems by X. Chen, Z.-C. Gu, and X.-G. Wen [Phys. Rev. B 83, 035107 (2011)]. It was found that such symmetry-protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking, and long-range entanglement (present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.
It is known that in certain nonbipartite quasi-one-dimensional spin systems in a magnetic field, in addition to the usual Pauli coupling of the spins to the field, new parity breaking three-spin interactions, i.e., chiral-spin interactions, are induced at higher order due to virtual processes involving the intrinsic electronic nature of the underlying spins. The strength of these interactions depends strongly on the orientation of the field, a feature that can be exploited to detect chiral effects experimentally. In many spin systems, these chiral interactions are generated and should be taken into account before any comparison with experiments can be made. We study the effect of the chiral interactions on certain quasi-one-dimensional gapped spin-12\frac{1}{2}21​ systems and show that they can potentially alter the physics expected from the Pauli coupling alone. In particular, we demonstrate that these terms alter the universality class of the commensurate-incommensurate transition in spin tubes. More interestingly, in weakly coupled XX zigzag ladders, we find that the field-induced chiral term can close the singlet gap and drive a second-order transition in the nonmagnetic singlet sector, which then manifests itself as a two-component Luttinger liquidlike behavior in the spin-correlation functions. Finally, we discuss the relevance of our results to experiments.