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arXiv:1302.3138 · Physical Review Letters

Canonical Thermal Pure Quantum State

Sho Sugiura, Akira Shimizu

Published Jul 1, 2013• 230 citations
arXivHTMLPDF

Abstract

A thermal equilibrium state of a quantum many-body system can be represented by a typical pure state, which we call a thermal pure quantum (TPQ) state. We construct the canonical TPQ state, which corresponds to the canonical ensemble of the conventional statistical mechanics. It is related to the microcanonical TPQ state, which corresponds to the microcanonical ensemble, by simple analytic transformations. Both TPQ states give identical thermodynamic results, if both ensembles do, in the thermodynamic limit. The TPQ states corresponding to other ensembles can also be constructed. We have thus established the TPQ formulation of statistical mechanics, according to which all quantities of statistical-mechanical interest are obtained from a single realization of any TPQ state. We also show that it has great advantages in practical applications. As an illustration, we study the spin-1/2 kagome Heisenberg antiferromagnet.

DOI
10.1103/physrevlett.111.010401

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A thermal equilibrium state of a quantum many-body system can be represented by a typical pure state, which we call a thermal pure quantum (TPQ) state. We construct the canonical TPQ state, which corresponds to the canonical ensemble of the conventional statistical mechanics. It is related to the microcanonical TPQ state, which corresponds to the microcanonical ensemble, by simple analytic transformations. Both TPQ states give identical thermodynamic results, if both ensembles do, in the thermodynamic limit. The TPQ states corresponding to other ensembles can also be constructed. We have thus established the TPQ formulation of statistical mechanics, according to which all quantities of statistical-mechanical interest are obtained from a single realization of any TPQ state. We also show that it has great advantages in practical applications. As an illustration, we study the spin-1/2 kagome Heisenberg antiferromagnet.
We formulate statistical mechanics based on a pure quantum state, which we call a ``thermal pure quantum (TPQ) state''. A single TPQ state gives not only equilibrium values of mechanical variables, such as magnetization and correlation functions, but also those of genuine thermodynamic variables and thermodynamic functions, such as entropy and free energy. Among many possible TPQ states, we discuss the canonical TPQ state, the TPQ state whose temperature is specified. % We also propose there are other TPQ states. In the TPQ formulation of statistical mechanics, thermal fluctuations are completely included in quantum-mechanical fluctuations. As a consequence, TPQ states have much larger quantum entanglement than the equilibrium density operators of the ensemble formulation. We also show that the TPQ formulation is very useful in practical computations, by applying the formulation to a frustrated two-dimensional quantum spin system.
We formulate statistical mechanics based on a pure quantum state, which we call a ``thermal pure quantum (TPQ) state''. A single TPQ state gives not only equilibrium values of mechanical variables, such as magnetization and correlation functions, but also those of genuine thermodynamic variables and thermodynamic functions, such as entropy and free energy. Among many possible TPQ states, we discuss the canonical TPQ state, the TPQ state whose temperature is specified. % We also propose there are other TPQ states. In the TPQ formulation of statistical mechanics, thermal fluctuations are completely included in quantum-mechanical fluctuations. As a consequence, TPQ states have much larger quantum entanglement than the equilibrium density operators of the ensemble formulation. We also show that the TPQ formulation is very useful in practical computations, by applying the formulation to a frustrated two-dimensional quantum spin system.
Quantum statistical mechanics can be fomulated based on a single pure quantum state.In this thesis, I establish this formulation, see the properties of such a pure quantum state, and apply the formulation to a practical application.Historically, it has been found that quantum states have robust "typicality", which is the property that almost all the realizable state give equilibrium values correctly for all "mechanical variables", such as energy and a correlation function.I generally call the pure quantum states which can represent the equilibrium state "thermal pure quantum (TPQ) state".I extend the typicality so that it is applicable to "genuine thermodynamic variables", such as entropy and temperature.I introduce many kinds of the TPQ states which include a "(grand)canonical TPQ state" , which is specified by intensive variables such as temperature, and a "microcanonical TPQ state", which is easy to generate.Owing to these results, I establish the new formulation of statistical mechanics based on the TPQ states.
We generalize the thermal pure quantum (TPQ) formulation of statistical mechanics, in such a way that it is applicable to systems whose Hilbert space is infinite dimensional. Assuming particle systems, we construct the grand-canonical TPQ (gTPQ) state, which is the counterpart of the grand-canonical Gibbs state of the ensemble formulation. A single realization of the gTPQ state gives all quantities of statistical-mechanical interest, with exponentially small probability of error. This formulation not only sheds new light on quantum statistical mechanics but also is useful for practical computations. As an illustration, we apply it to the Hubbard model, on a one-dimensional (1D) chain and on a two-dimensional (2D) triangular lattice. For the 1D chain, our results agree well with the exact solutions over wide ranges of temperature, chemical potential, and the on-site interaction. For the 2D triangular lattice, for which exact results are unknown, we obtain reliable results over a wide range of temperature. We also find that finite-size effects are much smaller in the gTPQ state than in the canonical TPQ state. This also shows that in the ensemble formulation the grand-canonical Gibbs state of a finite-size system simulates an infinite system much better than the canonical Gibbs state.
We propose a quantum algorithm to investigate finite-temperature properties of quantum many-body systems by implementing the microcanonical thermal pure quantum (TPQ) states [S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012)] on a quantum computer. In the formalism, the TPQ state is constructed by applying the power of a rescaled Hamiltonian to a random state. We clarify that, using techniques based on quantum signal processing, the power operation can be efficiently realized when the ground-state energy of a quantum system of interest is given with some precision. Our algorithm has an advantage in calculating thermodynamic quantities at low temperatures.