Let a general quantum many-body system at a low temperature adiabatically cross through the vicinity of the system’s quantum critical point. We show that the system’s temperature is significantly suppressed due to both the entropy majorization theorem in quantum information science and the entropy conservation law in reversible adiabatic processes. We take the one-dimensional transverse-field Ising model and the spinless fermion system as concrete examples to show that the inverse temperature might become divergent around the systems’ critical points. Since the temperature is a measurable quantity in experiments, it can be used, via reversible adiabatic processes at low temperatures, to detect quantum phase transitions in the perspectives of quantum information science and quantum statistical mechanics.Let a general quantum many-body system at a low temperature adiabatically cross through the vicinity of the system’s quantum critical point. We show that the system’s temperature is significantly suppressed due to both the entropy majorization theorem in quantum information science and the entropy conservation law in reversible adiabatic processes. We take the one-dimensional transverse-field Ising model and the spinless fermion system as concrete examples to show that the inverse temperature might become divergent around the systems’ critical points. Since the temperature is a measurable quantity in experiments, it can be used, via reversible adiabatic processes at low temperatures, to detect quantum phase transitions in the perspectives of quantum information science and quantum statistical mechanics.

arXiv:0908.3783Frontiers of PhysicsAug 13, 20112 citations

Renormalization group flows and quantum phase transitions: fidelity versus entanglement

Huan-Qiang Zhou

We compare the roles of fidelity and entanglement in characterizing renormalization group flows and quantum phase transitions. It turns out that the scaling parameter extracted from fidelity for different ground states succeeds to capture nontrivial information including stable and unstable fixed points, whereas the von Neumann entropy as a bipartite entanglement measure (or equivalently, majorization relations satisfied by the spectra of the reduced density matrix along renormalization group flows) often fails, as far as the intrinsic irreversibility-information loss along renormalization group flows-is concerned. We also clarify an intimate connection between the von Neumman entropy, majorization relations, and fidelity. The relevance to Zamolodchikov's c theorem is indicated.We compare the roles of fidelity and entanglement in characterizing renormalization group flows and quantum phase transitions. It turns out that the scaling parameter extracted from fidelity for different ground states succeeds to capture nontrivial information including stable and unstable fixed points, whereas the von Neumann entropy as a bipartite entanglement measure (or equivalently, majorization relations satisfied by the spectra of the reduced density matrix along renormalization group flows) often fails, as far as the intrinsic irreversibility-information loss along renormalization group flows-is concerned. We also clarify an intimate connection between the von Neumman entropy, majorization relations, and fidelity. The relevance to Zamolodchikov's c theorem is indicated.

arXiv:0704.2945Apr 23, 200710 citations

Phase Transitions in Quantum Systems

WORLD SCIENTIFIC eBooksAug 1, 20040 citations

Some basic aspects of quantum phase transitions

Diana V Shopova, Dimo I Uzunov

arXiv:cond-mat/0301423Physics ReportsApr 23, 200331 citations

Adiabatic preparation without quantum phase transitions

Gernot Schaller

Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians ${H}_{\mathrm{I}}$ and ${H}_{\mathrm{F}}$ . For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of ${H}_{\mathrm{F}}$ from the ground state of ${H}_{\mathrm{I}}$ . The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $\ensuremath{\tau}$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only nonadiabatic but also thermal excitations are likely to occur. It is shown that---using only local Hamiltonians---for the one-dimensional quantum Ising model and the cluster model in a transverse field the conventional straight-line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians ${H}_{\mathrm{I}}$ and ${H}_{\mathrm{F}}$. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of ${H}_{\mathrm{F}}$ from the ground state of ${H}_{\mathrm{I}}$. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $\ensuremath{\tau}$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only nonadiabatic but also thermal excitations are likely to occur. It is shown that---using only local Hamiltonians---for the one-dimensional quantum Ising model and the cluster model in a transverse field the conventional straight-line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.

arXiv:0807.2516Physical Review ASep 23, 200828 citations

{H}_{\mathrm{I}}

and

{H}_{\mathrm{F}}

. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of

{H}_{\mathrm{F}}

from the ground state of

{H}_{\mathrm{I}}

. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $\ensuremath{\tau}$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only nonadiabatic but also thermal excitations are likely to occur. It is shown that---using only local Hamiltonians---for the one-dimensional quantum Ising model and the cluster model in a transverse field the conventional straight-line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.