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arXiv:1705.08890 · Physical review. B./Physical review. B

Ferromagnetic transition in a one-dimensional spin-orbit-coupled metal and its mapping to a critical point in smectic liquid crystals

Vladyslav Kozii, Jonathan Ruhman, Liang Fu, Leo Radzihovsky

Published Sep 18, 2017• 12 citations
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Abstract

We study the quantum phase transition between a paramagnetic and ferromagnetic metal in the presence of Rashba spin-orbit coupling in one dimension. Using bosonization, we analyze the transition by means of renormalization group, controlled by an $\ensuremath{\varepsilon}$ expansion around the upper critical dimension of two. We show that the presence of Rashba spin-orbit coupling allows for a new nonlinear term in the bosonized action, which generically leads to a fluctuation driven first-order transition. We further demonstrate that the Euclidean action of this system maps onto a classical smectic-A--C phase transition in a magnetic field in two dimensions. We show that the smectic transition is second order and is controlled by a new critical point.

DOI
10.1103/physrevb.96.094419

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Publisher's Note: Ferromagnetic transition in a one-dimensional spin-orbit-coupled metal and its mapping to a critical point in smectic liquid crystals [Phys. Rev. B 96 , 094419 (2017)]

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Received 6 March 2018DOI:https://doi.org/10.1103/PhysRevB.97.099901©2018 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasFerromagnetismLiquid crystal phase transitionsQuantum phase transitionsPhysical SystemsQuantum wiresSmectic liquid crystalsTechniquesBosonizationGreen's function methodsPath integralsQuantum field theoryRenormalization groupPolymers & Soft MatterCondensed Matter, Materials & Applied Physics

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We study the quantum phase transition between a paramagnetic and ferromagnetic metal in the presence of Rashba spin-orbit coupling in one dimension. Using bosonization, we analyze the transition by means of renormalization group, controlled by an \ensuremathε\ensuremath{\varepsilon}\ensuremathε expansion around the upper critical dimension of two. We show that the presence of Rashba spin-orbit coupling allows for a new nonlinear term in the bosonized action, which generically leads to a fluctuation driven first-order transition. We further demonstrate that the Euclidean action of this system maps onto a classical smectic-A--C phase transition in a magnetic field in two dimensions. We show that the smectic transition is second order and is controlled by a new critical point.
We study the quantum phase transition between a paramagnetic and ferromagnetic metal in the presence of Rashba spin-orbit coupling in one dimension. Using bosonization, we analyze the transition by means of renormalization group, controlled by an ɛ expansion around the upper critical dimension of two. We show that the presence of Rashba spin-orbit coupling allows for a new nonlinear term in the bosonized action, which generically leads to a fluctuation driven first-order transition. We further demonstrate that the Euclidean action of this system maps onto a classical smectic-A–C phase transition in a magnetic field in two dimensions. We show that the smectic transition is second order and is controlled by a new critical point.
A long-range interaction via virtual particle-hole pairs between Fermi-liquid quasiparticles leads to a nonanalytic behavior of the spin susceptibility $\ensuremath{\chi}$ as a function of the temperature (T)(T)(T), magnetic field (B)(\mathbf{B})(B), and wave number. In this paper, we study the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of $\ensuremath{\chi}$ for a two-dimensional electron liquid. Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of ${\ensuremath{\chi}}_{zz}$ with TTT and ∣B∣|\mathbf{B}|∣B∣ saturates at the energy scale set by the SOI, that of ${\ensuremath{\chi}}_{xx}$ $(={\ensuremath{\chi}}_{yy})$ continues through this energy scale until renormalization of the electron-electron interaction in the Cooper channel becomes important. We show that the renormalization group flow in the Cooper channel has a nontrivial fixed point and study the consequences of this fixed point for the nonanalytic behavior of $\ensuremath{\chi}$. An immediate consequence of SOI-induced anisotropy in the nonanalytic behavior of $\ensuremath{\chi}$ is a possible instability of a second-order ferromagnetic quantum phase transition with respect to a first-order transition to an XYXYXY ferromagnetic state.
We discuss the quantum phase transition between a quantum nematic metallic state to an electron metallic smectic state in terms of an order-parameter theory coupled to fermionic quasiparticles. Both commensurate and incommensurate smectic (or stripe) cases are studied. Close to the quantum critical point (QCP), the spectrum of fluctuations of the nematic phase has low-energy ``fluctuating stripes.'' We study the quantum critical behavior and find evidence that, contrary to the classical case, the gauge-type of coupling between the nematic and smectic is irrelevant at this QCP. The collective modes of the electron smectic (or stripe) phase are also investigated. The effects of the low-energy bosonic modes on the fermionic quasiparticles are studied perturbatively, for both a model with full rotational symmetry and for a system with an underlying lattice, which has a discrete point group symmetry. We find that at the nematic-smectic critical point, due to the critical smectic fluctuations, the dynamics of the fermionic quasiparticles near several points on the Fermi surface, around which it is reconstructed, are not governed by a Landau Fermi liquid theory. Surprisingly, the quasiparticles in the smectic phase also exhibit non-Fermi liquid behavior. We also present a detailed analysis of the dynamical susceptibilities in the electron nematic phase close to this QCP (the fluctuating stripe regime) and in the electronic smectic phase.
In the presence of strong spin-independent interactions and spin-orbit coupling, we show that the spinor Bose liquid confined to one spatial dimension undergoes an interaction- or density-tuned quantum phase transition similar to one theoretically proposed for itinerant magnetic solid-state systems. The order parameter describes broken Z2Z_2Z2​ inversion symmetry, with the ordered phase accompanied by non-vanishing momentum which is generated by fluctuations of an emergent dynamical gauge field at the phase transition. This quantum phase transition has dynamical critical exponent z≃2z \simeq 2z≃2, typical of a Lifshitz transition, but is described by a nontrivial interacting fixed point. From direct numerical simulation of the microscopic model, we extract previously unknown critical exponents for this fixed point. Our model describes a realistic situation of 1D ultracold atoms with Raman-induced spin-orbit coupling, establishing this system as a platform for studying exotic critical behavior of the Hertz-Millis type.
We study the effect of spin-orbit coupling on both the zero-temperature and nonzero-temperature behavior of a two-dimensional Fermi gas. We include a generic combination of Rashba and Dresselhaus terms into the system Hamiltonian, which allows us to study both the experimentally relevant equal-Rashba-Dresselhaus (ERD) limit and the Rashba-only (RO) limit. At zero temperature, we derive the phase diagram as a function of the two-body binding energy and Zeeman field. In the ERD case, this phase diagram reveals several topologically distinct uniform superfluid phases, classified according to the nodal structure of the quasiparticle excitation energies. Furthermore, we use a momentum-dependent SU(2) rotation to transform the system into a generalized helicity basis, revealing that spin-orbit coupling induces a triplet pairing component of the order parameter. At nonzero temperature, we study the Berezinskii-Kosterlitz-Thouless (BKT) phase transition by including phase fluctuations of the order parameter up to second order. We show that the superfluid density becomes anisotropic due to the presence of spin-orbit coupling (except in the RO case). This leads both to elliptic vortices and antivortices, and to anisotropic sound velocities. The latter prove to be sensitive to quantum phase transitions between topologically distinct phases. We show further that at a fixed nonzero Zeeman field, the BKT critical temperature is increased by the presence of ERD spin-orbit coupling. Subsequently, we demonstrate that the Clogston limit becomes infinite: TBKT{T}_{\mathrm{BKT}}TBKT​ remains nonzero at all finite values of the Zeeman field. We conclude by extending the quantum phase transition lines to nonzero temperature, using the nodal structure of the quasiparticle spectrum, thus connecting the BKT critical temperature with the zero-temperature results.