[Paper Review] Slave rotor theory of antiferromagnetic Hubbard model

Ki‐Seok Kim, Jung Hoon Han|arXiv (Cornell University)|May 10, 2006

Physics of Superconductivity and Magnetism3 citations

TL;DR

This paper extends the slave-rotor mean-field theory to the antiferromagnetic phase of the Hubbard model by introducing a second rotor field to track spin degrees of freedom. It reveals a zero-temperature incoherent-to-coherent crossover within a finite window of Hubbard interaction strength, challenging the Hartree-Fock picture of fully coherent quasiparticles in the magnetic phase.

ABSTRACT

The slave-rotor mean-field theory of Florens and Georges is generalized to the antiferromagnetic phase of the Hubbard model. An effective action consisting of a spin rotor and a fermion is derived and the corresponding saddle-point action is analyzed. Zero-temperature phase diagram of the antiferromagnetic Hubbard model is presented. While the magnetic phase persists for all values of the Hubbard interaction U, the single-particle spectral function exhibits a crossover into an incoherent phase when the magnetic moment m (and the corresponding U values) lies within a certain window m_c < m < 1-m_c, indicating a possible deviation from the Hartree-Fock theory.

Motivation & Objective

• To extend the slave-rotor mean-field theory to describe the antiferromagnetic phase of the Hubbard model.
• To investigate the nature of quasiparticle coherence in the magnetic phase beyond Hartree-Fock theory.
• To analyze the role of gauge fluctuations in stabilizing a crossover between coherent and incoherent phases.
• To determine whether the magnetic order persists across all interaction strengths in the extended slave-rotor framework.
• To compare the mean-field results with dynamical mean-field theory and identify deviations from standard quasiparticle behavior.

Proposed method

• Introduce a dual rotor representation: $ c_{i\sigma} = e^{-i\theta_i - i\sigma\phi_i} f_{i\sigma} $, where $ \theta_i $ tracks charge and $ \phi_i $ tracks spin.
• Derive an effective action with U(1) gauge fields for charge and spin, incorporating constraints via Lagrange multipliers.
• Perform saddle-point analysis on the effective action to obtain self-consistent equations for $ \alpha, \beta, m, q $, with $ m $ as the magnetic moment.
• Identify Bose condensation (coherent phase) when $ q = \alpha D $, signaling a Higgs phase for the spin rotor.
• Analyze the stability of the mean-field solution against compact U(1) gauge fluctuations using Fradkin-Shenker instanton arguments.
• Determine the phase boundary for the incoherent phase via the condition $ \Delta_b = \sqrt{q^2 - (\alpha D)^2} > 0 $, with finite gap indicating incoherence.

Experimental results

Research questions

• RQ1 Does the slave-rotor formalism predict a breakdown of quasiparticle coherence in the antiferromagnetic phase of the Hubbard model?
• RQ2 What is the role of spin rotor fluctuations in determining the coherence of quasiparticles at zero temperature?
• RQ3 How does the inclusion of spin degrees of freedom via a second rotor field alter the phase diagram compared to the original slave-rotor theory?
• RQ4 Is the magnetic order parameter stable against gauge fluctuations in the extended slave-rotor framework?
• RQ5 Can the mean-field phase transition between coherent and incoherent phases be elevated to a crossover due to gauge fluctuations?

Key findings

• The antiferromagnetic phase persists for all values of the Hubbard interaction $ U $, consistent with Hartree-Fock theory.
• A window of incoherent quasiparticles emerges for intermediate $ U/D $, specifically when $ m_c \lesssim m \lesssim 1 - m_c $ with $ m_c \approx 0.8 $, indicating a crossover from coherent to incoherent behavior.
• The incoherent phase is characterized by a finite bosonic gap $ \Delta_b = \sqrt{q^2 - (\alpha D)^2} > 0 $, which suppresses coherent electron spectral weight.
• The Bose condensation condition $ q = \alpha D $ occurs at $ (U/D)_c \approx 0.81 $, marking the onset of the coherent phase.
• Gauge fluctuations from the compact U(1) spin gauge field $ a_{ij} $ convert the mean-field second-order transition into a crossover, as per Fradkin-Shenker instanton analysis.
• The magnetic order parameter remains gauge-invariant and unaffected by fluctuations, preserving long-range antiferromagnetic order across all $ U $.