Symmetry-protected topological phases at finite temperature

Viyuela, Oscar; Rivas, Ana María Rivas; Martín-Delgado, M. A.

doi:10.1088/2053-1583/2/3/034006

Cited by 47 publications

(63 citation statements)

References 109 publications

(210 reference statements)

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“…We actually made use of this when writing the current kernel in the form (30). Importantly, this redundancy is independent of the physical gauge freedom 16 and need not be considered further until we discuss the FCS approach [cf. Eq.…”

Section: B Gauge Freedom In Pumped Observablementioning

confidence: 99%

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Gauge freedom in observables and Landsberg's nonadiabatic geometric phase: Pumping spectroscopy of interacting open quantum systems

2017

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We set up a general density-operator approach to geometric steady-state pumping through slowly driven open quantum systems. This approach applies to strongly interacting systems that are weakly coupled to multiple reservoirs at high temperature, illustrated by an Anderson quantum dot. Pumping gives rise to a nonadiabatic geometric phase that can be described by a framework originally developed for classical dissipative systems by Landsberg. This geometric phase is accumulated by the transported observable (charge, spin, energy) and not by the quantum state. It thus differs radically from the adiabatic Berry-Simon phase, even when generalizing it to mixed states, following Sarandy and Lidar.As a key feature, our geometric formulation of pumping stays close to a direct physical intuition (i) by tying gauge transformations to calibration of the meter registering the transported observable and (ii) by deriving a geometric connection from a driving-frequency expansion of the current. Furthermore, our approach provides a systematic and efficient way to compute the geometric pumping of various observables, including charge, spin, energy and heat. These insights seem to be generalizable beyond the present paper's working assumptions (e.g., Born-Markov limit) to more general opensystem evolutions involving memory and strong-coupling effects due to low temperature reservoirs as well.Our geometric curvature formula reveals a general experimental scheme for performing geometric transport spectroscopy that enhances standard nonlinear spectroscopies based on measurements for static parameters. We indicate measurement strategies for separating the useful geometric pumping contribution to transport from nongeometric effects.A large part of the paper is devoted to an explicit comparison with the Sinitsyn-Nemenmann fullcounting statistics (FCS) approach to geometric pumping, restricting attention to the first moments of the pumped observable. Covering all key aspects, gauge freedom, pumping connection, curvature, and gap condition, we argue that our approach is physically more transparent and, importantly, simpler for practical calculations. In particular, this comparison allows us to clarify how in the FCS approach an "adiabatic" approximation leads to a manifestly nonadiabatic result involving a finite retardation time of the response to parameter driving. PACS numbers: 73.63.Kv, 05.60.Gg, 72.10.Bg 03.65.VfTABLE I. Comparison of geometric density-operator approaches relevant to this paper. Approach Prior works Present work (I) Adiabatic state Adiabatic mixed-state geometric phase 37,38 Zero geometric phase for adiabatic steady-state [Sec. III A] evolution (ASE) Mixed-state adiabatic-response correction 37,38 Zero geometric phase for nonadiabatic state [Sec. III A] Gauge freedom related to eigenvector rescaling 37,38 Restriction of gauge freedom by normalization and hermiticity [Sec. III A] (II) Full counting Geometric part of generating function 40,44,72-75 Restriction of gauge freedom by real-valuedness observable [Sec. V C 3] ...

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Section: B Gauge Freedom In Pumped Observablementioning

confidence: 99%

“…II C). These technical assumptions 16 The rewriting of the result (30) in Appendix A involves adding a commutator [X r ,•] to the expression, which is invariant under the physical gauge transformations (57). This means one can do such rewriting at any stage of the calculation.…”

Section: R Gmentioning

confidence: 99%

Gauge freedom in observables and Landsberg's nonadiabatic geometric phase: Pumping spectroscopy of interacting open quantum systems

2017

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“…(1) to pure states. Nevertheless, one can still verify that, with extra care, A U can approach the corresponding Berry connection as T → 0 16,25 . A more severe problem of the Uhlmann connection is that the density matrix bundle is topologically trivial.…”

Section: Challenge Of Extracting Topological Properties Of Mixed mentioning

confidence: 97%

“…This means that A U is a global section of the U (n) bundle, which then implies that this bundle is topologically trivial. As a consequence, all characteristic classes, such as the Chern class and Chern character, are all zero 25 . This implies that even though the Uhlmann connection approaches the Berry connection as T → 0, the underlying bundle structures can still be different.…”

Section: Challenge Of Extracting Topological Properties Of Mixed mentioning

confidence: 99%

Thermal Uhlmann-Chern number from the Uhlmann connection for extracting topological properties of mixed states

Guo

Chien

2018

Phys. Rev. B

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The Berry phase is a geometric phase of a pure state when the system is adiabatically transported along a loop in its parameter space. The concept of geometric phase has been generalized to mixed states by the so called Uhlmann phase. However, the Uhlmann phase is constructed from the Uhlmann connection that possesses a well defined global section. This property implies that the Uhlmann connection is topologically trivial and as a consequence, the corresponding Chern character vanishes. We propose modified Chern character whose integral gives the thermal Uhlmann Chern number, which is related to the winding number of the mapping defined by the Hamiltonian. Therefore, the thermal Uhlmann Chern number reflects the topological properties of the underlying Hamiltonian of a mixed state. By including the temperature dependence in the volume integral, we also introduce the non-topological thermal Uhlmann Chern number which varies with temperature but is not quantized at finite temperatures. We illustrate the applications to a two-band model and a degenerate four-band model.

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“…Identifying topological phases at finite temperature has received attention in its own right through the definition of the topological Uhlmann number [6], which generalizes the notion of a Chern number computed from the band structure to finite temperature. In the quantum Hall setting, approaches have also been developed to compute topological conductivities at finite temperature and in the presence of disorder [7,8].…”

Section: Introductionmentioning

confidence: 99%

Conformal energy currents on the edge of a topological superconductor

et al. 2017

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The boundary of a two-dimensional topological superconductor can be modeled by a conformal field theory. Here we demonstrate the behaviors of this high level description emerging from a microscopic model at finite temperatures. To achieve that, we analyze the low-energy sector of Kitaev's honeycomb lattice model and probe its energy current. We observe that the scaling of the energy current with temperature reveals the central charge of the conformal field theory, which is in agreement with the Chern number of the bulk. Importantly, these currents can discriminate between distinct topological phases at finite temperatures. We assess the resilience of this measurement of the central charge under coupling disorder, bulk dimerization, and defects at the boundary, thus establishing it as a favorable means of experimentally probing topological superconductors.

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Symmetry-protected topological phases at finite temperature

Cited by 47 publications

References 109 publications

Gauge freedom in observables and Landsberg's nonadiabatic geometric phase: Pumping spectroscopy of interacting open quantum systems

Gauge freedom in observables and Landsberg's nonadiabatic geometric phase: Pumping spectroscopy of interacting open quantum systems

Thermal Uhlmann-Chern number from the Uhlmann connection for extracting topological properties of mixed states

Conformal energy currents on the edge of a topological superconductor

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