[Paper Review] Slave-spin approach to the strongly correlated systems

Masoud Mardani, Mohammad-Sadegh Vaezi+1 more|arXiv (Cornell University)|Nov 25, 2011

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper introduces a novel slave-spin approach using spin-one particles to study strongly correlated systems, replacing the slave rotor with a more tractable spin-one degree of freedom that can be mapped via Schwinger bosons or fermions. The method enables a smooth connection to the non-interacting limit and correctly describes the Mott transition in the Kane-Mele-Hubbard model, identifying a gapped chiral spin liquid phase at moderate U/t₁ and small spin-orbit coupling t₂.

ABSTRACT

In this paper, we develop a new type of slave particle method which is similar to the slave rotor model except that the quantum rotor is substituted by a spin one slave particle. The spin-one slave particle itself can be represented in terms of Schwinger bosons/fermions. This approach is more conveniently applicable to the strongly correlated Hamiltonians with on-site Hubbard interaction and resolves the limitations of using the slave rotor model as well as the Anderson-Zou slave particle technique. For instance, the mean-field parameters of the slave spin method do not vanish above the Mott transition and this approach is smoothly connected to the non-interacting limit. As an example, we study the phase diagram of the Kane-Mele-Hubbard model using our current approach. In the absence of the spin-orbit interaction, the Mott transition occurs at $U_c ~ 3 t_1$. Several aspects of the slave spin method, its gauge theory and various possible mean-field states associated with this approach have been discussed.

Motivation & Objective

• To develop a new slave-particle method that overcomes limitations of the slave rotor and Anderson-Zou techniques in describing strongly correlated systems.
• To enable a smooth connection to the non-interacting limit, particularly at half-filling where previous methods fail.
• To provide a gauge-invariant formulation that correctly captures the Mott transition and topological phases in models like the Kane-Mele-Hubbard model.
• To show that the slave-spin approach naturally supports gapped spin liquid phases with non-zero Chern numbers, even in the presence of spin-orbit coupling.
• To demonstrate that the method remains valid above the Mott transition, where mean-field parameters do not vanish, unlike in the slave rotor approach.

Proposed method

• Decompose the electron operator as $ c_{i, ho}^{ ho} = S^{+}_{i} f_{i, ho}^{ ho} $, where $ S^{+}_{i} $ is a spin-one operator creating charge and $ f_{i, ho}^{ ho} $ are spinon operators carrying spin.
• Assign the charge sector via $ S^{z}_{i} = n_e(i) - 1 $, with $ S^z = +1 $ for doubly occupied, $ 0 $ for half-filled, and $ -1 $ for empty sites.
• Represent the spin-one operators using Schwinger bosons or fermions, enabling a tractable mean-field treatment of the slave-spin degrees of freedom.
• Construct a gauge theory with two U(1) gauge fields: $ U(1)_c $ for charge and $ U(1)_s $ for spin, with the slave-spin constraint enforced via Lagrange multipliers.
• Use the Schwinger boson representation to derive the effective action and analyze the gauge field dynamics, including Higgs and Chern-Simons terms.
• Apply the method to the Kane-Mele-Hubbard model, analyzing phase transitions via mean-field parameters and gauge symmetry breaking via condensates and Chern numbers.

Experimental results

Research questions

• RQ1 How can a slave-particle method be constructed that remains valid across the entire phase diagram, including above the Mott transition?
• RQ2 Can a spin-one slave particle provide a more stable and physically consistent description than the slave rotor or Anderson-Zou approaches?
• RQ3 What is the role of gauge symmetry breaking in stabilizing topological and gapped spin liquid phases in the Kane-Mele-Hubbard model?
• RQ4 How does the inclusion of spin-orbit coupling affect the emergence of chiral spin liquid and topological insulator phases?
• RQ5 Can the slave-spin method correctly describe the non-interacting limit and the semi-metal phase at weak coupling?

Key findings

• The slave-spin method is smoothly connected to the non-interacting limit, avoiding the spurious superconducting phase found in the Anderson-Zou approach at half-filling.
• The mean-field parameters do not vanish above the Mott transition, in contrast to the slave rotor model, ensuring applicability across the full phase diagram.
• For $ U < U_{c,1} hinspace ext{with} hinspace t_2 = 0 $, the system is in a semi-metal phase with gapless spinons and $ raket{b_{3,i}^{ ho}} eq 0 $, indicating $ U(1)_s $ gauge symmetry breaking.
• In the intermediate $ U $ range ($ U_{c,1} < U < U_{c,2} $), a gapped chiral spin liquid phase emerges with non-zero Chern number for spinons, stabilized by the $ U(1)_s $ gauge field via a Chern-Simons term.
• For $ U > U_{c,2} $, the spinons remain gapless, and instanton effects proliferate, leading to spontaneous symmetry breaking and phases like in-plane Néel order or valence bond solids.
• The method correctly identifies a topological band insulator phase at large $ U $, where spinons have opposite Chern numbers, resulting in helical edge states and a gapped bulk.