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arXiv:0704.3616 · Physical Review B

Entanglement and topological entropy of the toric code at finite temperature

Claudio Castelnovo, Claudio Chamon

Published Nov 29, 2007• 170 citations
arXivHTMLPDF

Abstract

We calculate exactly the von Neumann and topological entropies of the toric code as a function of system size and temperature. We do so for systems with infinite energy scale separation between magnetic and electric excitations, so that the magnetic closed loop structure is fully preserved while the electric loop structure is tampered with by thermally excited electric charges. We find that the entanglement entropy is a singular function of temperature and system size, and that the limit of zero temperature and the limit of infinite system size do not commute. The two orders of limit differ by a term that does not depend on the size of the boundary between the partitions of the system, but instead depends on the topology of the bipartition. From the entanglement entropy we obtain the topological entropy, which is shown to drop to half its zero-temperature value for any infinitesimal temperature in the thermodynamic limit, and remains constant as the temperature is further increased. Such discontinuous behavior is replaced by a smooth decreasing function in finite-size systems. If the separation of energy scales in the system is large but finite, we argue that our results hold at small enough temperature and finite system size, and a second drop in the topological entropy should occur as the temperature is raised so as to disrupt the magnetic loop structure by allowing the appearance of free magnetic charges. We discuss the scaling of these entropies as a function of system size, and how the quantum topological entropy is shaved off in this two-step process as a function of temperature and system size. We interpret our results as an indication that the underlying magnetic and electric closed loop structures contribute equally to the topological entropy (and therefore to the topological order) in the system. Since each loop structure per se is a classical object, we interpret the quantum topological order in our system as arising from the ability of the two structures to be superimposed and appear simultaneously.

DOI
10.1103/physrevb.76.184442

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We calculate exactly the von Neumann and topological entropies of the toric code as a function of system size and temperature. We do so for systems with infinite energy scale separation between magnetic and electric excitations, so that the magnetic closed loop structure is fully preserved while the electric loop structure is tampered with by thermally excited electric charges. We find that the entanglement entropy is a singular function of temperature and system size, and that the limit of zero temperature and the limit of infinite system size do not commute. The two orders of limit differ by a term that does not depend on the size of the boundary between the partitions of the system, but instead depends on the topology of the bipartition. From the entanglement entropy we obtain the topological entropy, which is shown to drop to half its zero-temperature value for any infinitesimal temperature in the thermodynamic limit, and remains constant as the temperature is further increased. Such discontinuous behavior is replaced by a smooth decreasing function in finite-size systems. If the separation of energy scales in the system is large but finite, we argue that our results hold at small enough temperature and finite system size, and a second drop in the topological entropy should occur as the temperature is raised so as to disrupt the magnetic loop structure by allowing the appearance of free magnetic charges. We discuss the scaling of these entropies as a function of system size, and how the quantum topological entropy is shaved off in this two-step process as a function of temperature and system size. We interpret our results as an indication that the underlying magnetic and electric closed loop structures contribute equally to the topological entropy (and therefore to the topological order) in the system. Since each loop structure per se is a classical object, we interpret the quantum topological order in our system as arising from the ability of the two structures to be superimposed and appear simultaneously.
We study quantum quenches in the two-dimensional Kitaev toric code model and compute exactly the time-dependent entanglement entropy of the nonequilibrium wave function evolving from a paramagnetic initial state with the toric code Hamiltonian. We find that the area law survives at all times. Adding disorder to the toric code couplings makes the entanglement entropy per unit boundary length saturate to disorder-independent values at long times and in the thermodynamic limit. There are order-one corrections to the area law from the corners in the subsystem boundary but the topological entropy remains zero at all times. We argue that breaking the integrability with a small magnetic field could change the area law to a volume scaling as expected of thermalized states but is not sufficient for forming topological entanglement due to the presence of an excess energy and a finite density of defects.
We study topological order in a toric code in three spatial dimensions or a 3+1D3+1\text{D}3+1D Z2{\mathbb{Z}}_{2}Z2​ gauge theory at finite temperature. We compute exactly the topological entropy of the system and show that it drops, for any infinitesimal temperature, to half its value at zero temperature. The remaining half of the entropy stays constant up to a critical temperature Tc{T}_{c}Tc​, dropping to zero above Tc{T}_{c}Tc​. These results show that topologically ordered phases exist at finite temperatures, and we give a simple interpretation of the order in terms of fluctuating strings and membranes and how thermally induced point defects affect these extended structures. Finally, we discuss the nature of the topological order at finite temperature and its quantum and classical aspects.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
We consider topological entanglement entropy (TEE) at finite temperature for CSS codes, which include some ordinary topological-ordered systems such as the toric code and some fracton models such as the Haah's code and the X-cube model. We find, under the assumption that there is no extended critical phase, the finite-temperature TEE is a piecewise constant function of the temperature, with possible discontinuities only at phase transitions. We then consider phase transitions of CSS codes. We claim that there must exist a phase transition at zero temperature for any CSS codes in 2D and 3D with topological order. This statement can be rigorous proved for some familiar examples, while for general models it can be argued based on the low-temperature expansion. This indicates the break down of topological orders at finite temperature. We also discuss possible connections with self-correcting quantum memory.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.